Which is flatter for a trip from Los Angeles to Kansas in August, I70 or I40?

You do not say what part of Los Angeles you will be leaving from.
To get to either the I40 or I70 from Los Angeles would require taking I15. While there are not very many sharp turns, you do have to go through the Cajon Pass, and the road is quite steep. If you want to continue on I15 to I70, there are steep hills at the California/Nevada border.
The other option might be to take I10 to Phoenix, then I17 north to the I40. Or, continue on the I10 to the I25 at Las Cruces.
You will have to check that route for the other requirements you list.

Note, there are other non-interstate options, depending on where in Los Angeles you want to leave from. If you are leaving from the San Fernando Valley, take SR 14 from the I5, get off at Sierra Highway and take it out north to Palmdale where it becomes Pearblossom Highway. Continue on to Highway 138, and take the 138 to the I15. You can also use highway 18 (Palmdale Road) between highway 138 and the I15.
I have used this route to avoid the evening rush-hour traffic on SR 14 going to Adelanto. A bit more of a distance, but a much more pleasant drive.
Sierra highway is only a two lane road through much of this route, but the hills are relatively gentle.

Addendum – if you do choose the I10/I25 route, do not try continuing on north on I25 from Albuquerque. The road to Santa Fe is quite steep, as there is a 2000 ft rise between Santa Fe and Albuquerque. There is also nearly a 1000 ft drop between Colorado Springs and Denver.

I would not count on having continuous cell service on either road. Out here they attempt to cover the highways but they don’t eliminate every last terrain shadow.

There’s also the Cajon pass to consider–it’s on I-15, not far from Los Angeles and either of your proposed routes go through it. From what you say you might be bothered by it, also.

Also, expect a substantial bridge when you cross the Colorado river. You didn’t mention bridges specifically but given your other fears I figured I should mention it.

For many people the Mojave Desert in California is the most difficult part of the trip. There are no mountains but you will have difficulty with gas, rest areas, and cell service. There are almost no services from Barstow, CA to Needles (I-40) or Barstow to Las Vegas (I-15/I-70). You’ll want to stop in Barstow to eat, fill your gas tank, and get some water. The Mojave Desert in August will be the hottest place in the country. If possible you will want to time your departure from Los Angeles so you are not going through the Mojave at the hottest part of the day.

Do not like heights… Wow, that’s not even a contest.

I-70 was such a difficult and perilous road to build that it wasn’t even finished until the 1990s. It goes right through the center of the Rocky mountains. Through Glenwood Canyon, I-70 was built on a ledge and does all sorts of crazy cantilevering out over the river. (You can see it from Amtrak’s California Zephyr). I swear they’re using anti-gravity repulsors to hold it up. So negative Ghost rider. Wave off, wave off!

I-40 is deadly dull. You remember the movie Cars, where the freeway bypassed all the towns along the 2-lane road, and Lightning gets off the very boring freeway and winds up in one of those towns. They were specifically referring to I-40.

That’s not really an insult. The freeways were designed to be deadly dull, because it reduces accidents. Although they prefer to call it "designed to a consistent engineering standard with modest grades, modest curves, and good sight lines (as far as seeing the road and other traffic), particularly on vertical curves (e.g. going over the top of a crest)". Even I-70 has this. It’s just that with I-70, to put the road where it needed to be to meet those standards, it is often suspended seemingly magically in the air.

(Although, one part of "dull" that’s "deadly" is drowsy driving. Utah is really big into warning people about that, but then, Utah has a freeway across the salt flats.)

This alone is the deal-killer for I-70. It’s a wonder of the world, but not for you. https://www.youtube.com/watch?v=LNqcNH7ez4k

I need safe accommodation and rest areas on the road.

"Rest areas" as you know them in the east, with the full service parking, restrooms, vending machines, picnic areas etc. can get pretty sparse in the American west. Some states put minimal "rest areas" at an exit, out on the regular roads, so they can use the same rest area for both directions. I’m not a fan; it’s too easy for people on foot to reach those and do crime.

All rest areas will be full of trucks all night. They are sleeping. The area will be full of the growl of diesel engines. These are the engines on refrigerated trailers to run the freezers inside them, or the trucker is running his diesel to run his cab A/C. That trick won’t work for you. Do not attempt to run your gas engine while you sleep, or you’ll wake up out of fuel or with a broken down engine!

Other than a decent full-service rest area, my top choice is any of the popular "Travel Centers" – Pilot, Flying J, TA, Petro, etc. They are perfectly fine and safe when they’re in the middle of nowhere. One side is for trucks, one side is for cars. Choose the "car" side obviously. These are open 24 hours and it is safe to sleep in their parking lots. They’re not quiet, but neither will be a rest area.

Those "Travel Centers" typically have a "7-11" type food store, a sit-down restaurant, and at least one fast-food offering better than "Subway" – McDonalds, Arby’s, Taco Bell, etc.

I need to refill gas very frequently.

You’ll find a travel center, or a town with a row of gas stations, at least every 100 miles (160 km). That will be more than often enough for any car. Use your smart phone, put the map area on the area 200 miles ahead of you, and type "Gas". It will show you where gas stations are in that area.

Your car will generally get much better fuel economy going 55 mph (90 km/h) on the freeway than it will driving around town. However, as your speed increases, your fuel economy will worsen again, until by 80 mph (130 km/h) it’s "worse" by city mileage.

You don’t have to drive 80 mph just because the speed limit sign says that. Many trucks are governor limited to 65 mph (105 km/h) and some even go 55-60. It saves fuel. Go the speed you are comfortable with. Know your "personal minimums" and be OK with that.

If fuel expense is a concern and you want to improve your fuel economy, here are my top strategies in order:

• Slow Down. The faster you go, the worse your MPG due to aerodynamic drag.
• Turn A/C off, put vent on maximum and recirculation on "outside fresh air". Keep the windows closed (they add so much aerodynamic drag that they’re worse than the A/C!) If it’s hot, it’ll be HOT. Hydrate like crazy – always have two 32-ounce (1 litre) drink cups in rotation, one with a drink, one with spare ice. Almost every travel center will let you fill your drink cups with ice for free if you ask nicely. You’ll want to stop every 90 minutes to re-fill ice and to pee – if you’re not peeing you’re dehydrating!
• If there’s no crosswind, follow trucks. Lurk about 100′ (7 carlengths, 2 truck lengths, 30m) behind. This causes a "drafting" effect if there’s no crosswind. (yes, this works at 100′. I’ve played with this extensively with a car with instant MPG reading. NEVER tailgate!)

These combine nicely. If you’re uncomfortable going faster than 65 mph, find a truck going 65 mph and follow it.

Cellphone signal is very important.

Check your cell coverage map. Generally AT&T and Verizon make an effort to cover every freeway and much of rural America.

Many cell phone providers (e.g. metroPCS) only market to urban customers, and only bother to maintain a cell network in cities. When one of their customers goes out into the country, either it doesn’t work, or they rely on a process called called roaming – that is, borrowing the network of typically AT&T or Verizon, and that means your cell carrier is paying them for your usage. As such, the carrier rations and limits the roaming permitted. For instance they may allow voice/TXT roaming but not data roaming, or may squeeze data roaming speeds down to "check email" speeds. So have the "roaming" conversation with your cell provider if it isn’t AT&T or Verizon.

Also keep in mind that they just don’t have 5G (or even 4G) out in the sticks. It’s not a deployment priority since there are so few users and the airspace isn’t crowded (the main problem 4G/5G aim to solve).

Note it is both illegal and dangerous to be using a smart-phone or tablet while driving. When you are cruising at 80 mph, loss of control due to lapse of attention is much more serious! Further, the truckers have CB radios, and will get on channel 19 or 9 and tell the cops about any distracted drivers.

NTSB’s position on any electronics in the driver’s seat is that nothing should be used except for things designed to support the driving task. So, basically, maps.

While I-70 does pass through the Rocky Mountains, it is an Interstate Highway and not a "mountain road."

For instance, here’s a random location near Vail CO, facing westbound. The eastbound lanes are to the left of the concrete center median.

(image cribbed from Google Street View)

I need to refill gas very frequently.

That’s generally not a problem on US Interstates. The longest distance between gas stations is on I-70 in Utah with 106 miles. I believe any car can do this, and any Yaris should be good for at least 300 miles

I am afraid of heights and can’t drive on mountain roads.

I-70 has the highest elevation of any US Interstate and peaks at 11,158 ft . I-40 goes only up to 7,320 ft. This being said, both a perfectly drivable Interstate highways and not curvy and narrow mountain roads.

I need safe accommodation and rest areas on the road.

Any Interstate rest stop is well equipped and quite safe. If you are worried about accommodation, just book a hotel from a major chain up-front and stay there. Something like a Holiday Inn, Marriot Court Yard, Hampton Inn, etc.

cellphone signal is very important.

Check your carriers coverage map. Coverage is not the same for all carriers. Chances are coverage will be quite good on the Interstate other than maybe a few dead spots along the way.

Both routes are perfectly feasible. I-40 seems better for your specific requirements.

Which is flatter for a trip from Los Angeles to Kansas in August, I70 or I40?

Based on elevation, that’s clearly I-40.

With all other things being equal(1), I-70 runs through the rockies , while I-40 runs south of them. So based on your desire not to travel on mountain roads that would make I-40 the better choice.

I can’t attest to what I-40 is like between LA and Flagstaff, AZ, but east of Flagstaff the geography is pretty flat with occasional small mountains on either side. Once you get past Albuquerque, NM, it’s very flat all the way to Oklahoma City, OK.

So I think your preferred route is via I-40 to Oklahoma City and then head north on I-35 to Wichita. You could also pick up route 54 from Tucumcari, NM directly to Wichita.

Finally, I just looked at Google, LA to Wichita.

1. LA to Tucumcari to Wichita. 1379 miles, 21 hours
2. LA to OKC to Wichita. 1487 miles, 21 hours 45 minutes
3. LA to Denver to Wichita. 1533 miles, 22 hours.

So the southern route via I-40 is also the shortest by a small degree.

FWIW there are extensive wikipedia articles on the interstates: Interstate 70 and Interstate 40. So you can review the routes and get better acquainted with your trip. Also each of those links points to a more detailed description for each state that they cross.

(1) Gas and accommodation are found reasonably easy on I-40 in towns. You know the range of your car, so you can plan driving legs accordingly in order to minimize range anxiety.

You should also have cell service along I-40 as well.

Don’t expect to find fully equipped rest areas outside of towns. You more likely just to find an area to pull off the freeway that has zero services. The exception being welcome centers at/near each state’s border.

Finally, while I can’t say anything about other cities, I would not recommend staying overnight in Albuquerque due to theft from vehicles (or on occasion entire trailers taken from vehicles). I state this from living in the area and following the local news.

Finally, finally. Based on John’s comment about roadworks, each state’s DOT (or equivalent) has a website listing current road conditions (such as road works and accidents). So I would bookmark those sites and refer to them during your trip so at least you will know where to expect delays.